"We're one of seven nations in the entire world that allow abortion on demand at 20 weeks, the fifth month of the pregnancy. I'd like to get us out of that club," the South Carolina Republican, who is running for president, said Monday.

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Other countries that don't ban elective abortions after 20 weeks include Canada, China, the Netherlands, North Korea, Singapore and Vietnam.

Graham's remarks come as the Senate is expected to take a procedural vote on Tuesday on proceeding to a House-passed bill that would ban most abortions after 20 weeks.

The South Carolina senator, who has introduced a similar bill in the Senate that has gained 45 co-sponsors, said that he is "honored to lead this fight."

Acknowledging that the bill would likely be blocked from being taken up, Graham asked Democrats what they're "afraid of?"

"I'm not afraid of my idea," he added. "I'm very disappointed that we can't even have the debate. This, I promise you, as long as I'm here and many others on our side and hopefully some of the side over here [Democrats], this debate will continue until we get the right answer."

Democrats have, separately, introduced legislation that would roll back certain state-level restrictions on abortions. Graham said he would back allowing Democrats to offer their legislation as an amendment to the abortion ban legislation.

The move, however, would be largely symbolic, with Democrats likely not having the votes needed.

The Senate's fight over the abortion bill comes as Republican leadership is under pressure to allow a vote on defunding Planned Parenthood in the wake of a series of controversial videos regarding its practices on fetal tissue.

Graham said those videos have "woken America up" to the debate over late-term abortions.

"This discussion about harvesting organs from children late in the birthing process has woken America up," he added. "So I promise you, this is a debate worthy of this body, worthy of this country, one that we're going to have over and over again until we can get a vote, and I'm not going to stop."